Urs Schneider's allegations about the Ashtekar Shadow states paper

Urs has made serious allegations about the paper
"Quantum gravity, shadow states, and quantum mechanics"
by Abhay Ashtekar, Stephen Fairhurst, and Joshua Willis.

However I've not seen any place where he openly refers to the paper by title and named the authors! So I guess there have been vague hints and rumors and suggestions flying around, with no obvious attachment to substance. Several people besides myself seem to have been confused about what was being discussed.

I saw a stray reference to a license plate (gr-qc/0207106) and mistook it for some other 2002 paper, in the absence of identifying context, and Urs corrected me and said it was by Ashtekar. So then I knew. I dont ordinarily recognize papers by their arxiv numbers or my friend's cars by their license plates

Anyway, when you have serious criticisms of a particular paper, then, as Urs would doubtless be the first to point out, it is good scholarly practice to be open and explicit and public about it. So I have started a Ashtekar "Shadow states" thread and I hope Urs will post there and say exactly what he thinks is wrong or dubious about the paper.

Criticism is often extremely valuable and useful. My personal estimation of Urs is very high. I have the highest regard for his intelligence and expertise. Also a high regard for Ashtekar--who rates a thread of his own instead of being hidden in some closet of a Thomas Thiemann thread.

So I think it will most probably be very helpful to have all of Urs objections to this paper to be made clear and out in the open.

There is a particular reason why I think this could be especially interesting to see in the present, which I will explain in the next post.

Urs,
One reason I think it would be helpful to have you spell out your criticisms is that currently there are several different approaches being tried to connect with the low energy limit in LQG.

Ashtekar et al's paper is one of these initiatives. When a number of different tentative attempts are being made criticism can help to eliminate the less-promising ones and the false starts.

Indeed I was under the impression that this paper had followed a blind alley, but someone corrected me about this. I had that impression first because it does not look and feel like the LQG I'm used to: didn't seem to be typical Loop Gravity----and second because it has gotten almost no citations compared with what major Ashtekar papers usually get. But those were just impressions, not based on anything solid. Another poster here (who I believe knows more than I do) has told me the paper is relevant to LQG!

So here we have an important research area (the different avenues people are exploring to connect with physics at low energy) and a paper that is potentially important (in at least one knowledgeable person's view) and you have made analysis that apparently casts doubt on it. Something "odd" or "weird" or "non-standard" or "drastically different" I seem to recall.

Though I think you said you found no flaw in the mathematics per se.

Hope I am not misquoting your objections to the paper. I, and maybe others besides myself, would be happy if you would discuss this in the regular way, in a thread explicitly devoted to this particular paper.

In Urs' defense, I happen to know that he will be away for the next three days enjoying a nice internet-free vacation on the beach.

I think that instead of insinuating that Urs is being somehow academically underhanded by not provided you the references AGAIN you should be down on your hands and knees thanking him for his patience to even talk to you. He went through a LOT of trouble to find those page numbers and references in Rovelli's book for you the first time. I specifically remember him telling you, but I can't seem to find them again either. Now you want him to dig through Rovelli's book AGAIN to find those SAME equations and page numbers that he already went to the trouble of finding for you himself. He is extremely busy with his OWN research. In case you didn't know, graduate school in physics is not a stroll in the park. I hope he enjoys his much needed break away from studies.

In Urs' defense, I happen to know that he will be away for the next three days enjoying a nice internet-free vacation on the beach.

...

That is great. I think he's a nice person too. Hope he enjoys the break.

Now you want him to dig through Rovelli's book AGAIN to find those SAME equations and page numbers that he already went to the trouble of finding for you himself.
...

Eric,

that is not what I am asking here
this is about a paper by Ashtekar et al
that Urs says he has discussed at length

Urs says: "Since I have mentioned this paper for quite a while now and many times, here, on s.p.r. and at the Coffee Table, have written summaries and critical discussions of this paper in these three groups, have compared its techniques to those used by Thomas Thiemann,..."

I cant find his discussion in the Thiemann thread
or any place where it is made explicit what paper he's talking about
I would like it to be in a place we can find it
with the name of the paper being discussed
so its clear what is being talked about

Do you have any links to discussion of this paper?

Sorry if I seem demanding but Ashtekar is an important
person---one of the founders of the field---and critical
discussion of one of his papers really needs its own thread.
I dont want to argue with Urs claims! I dont want them buried
somewhere in the TT thread or on some other board.
So far all I see is conclusion-words like "weird" "drastically different" "not honest" (meaning not genuine).

If work by these authors (Ashtekar, Willis, Fairhurst) is being
put on trial and judged dubious or off-beat or inconsistent with commonsense or whatever (I dont know exactly what the accusations amount to), it needs to be out in the open.

I found another place where Urs was talking about the
Ashtekar/Fairhurst/Willis paper, but the title of the paper was no where in sight. And none of the authors were mentioned.
there was just the arxiv code--the gr-qc number 0207106

Now the great thing is this: not only is Urs obviously a nice, fairminded, and intelligent person but he could also be correct in his assessment of Ashtekar/Fairhurst/Willis's paper.
It could be that the paper is drastically (as Urs says) speculative and even dangerously abnormal. We only need to make this claim clearly and openly and clear so people can worry about that if they want. Here is what he was saying on that occasion about

---------quote from Urs post----
And let me emphasize that by 'highly speculative' I mean something drastic. Of course every theory of quantum gravity in the absence of experiments has to be speculative. In string theory there is the single and obvious speculation that strings exist. Everything else follows. If they don't exist, they don't. Fine.

But in LQG the speculation is that at the Planck scale the quantum principle itself is radically different from everything we know so far. Maybe one can argue that the modified priciple should still be called 'canonical'. Words are arbitrary. But it still refers to a concept drastically different from what is usually called canonical, outside the LQG-literature. You wouldn't claim that the LQG-like quantization of the 1d nonrelativistic particle in gr-qc/0207106 is 'canonical' would you? It's not canonical - it's weird!

I can say that with full confidence because if we know one thing for sure it is how the quantum theory of the 1d nonrel particle works. And it works very differently from the supposedly 'canonical' theory that is presented in gr-qc/0207106....
--------end quote---------

Marcus, Urs discusses the shadow states paper, with the title and the authors' names, plus a link, at this permalink on the Coffee Table site. As he said, he also discussed it at s.p.r., but that was just a summary of his discussion on the Coffee Table. It has page numbers and all.

Originally posted by selfAdjoint Marcus, Urs discusses the shadow states paper, with the title and the authors' names, plus a link, at this permalink on the Coffee Table site...

Great! Thanks. I will immediately paste it in here. I was put off when he refused to give me a link like that. Probably many here have already read this, but for the convenience of any new visitors, and my own (since I would have to modify my software to make Coffee Table legible), here it is:

This morning I continued reading A. Ashtekar, S. Fairhurst and J. Willis, Quantum gravity, shadow states and quantum mechanics, 2002.

According to Demain Cho’s comments here this paper should contain the key peculiarities of LQG-like quantization in a tractable toy example.

The basic idea is to see what happens when the correspondence principle of elementary quantum mechanics is violated. Taking the risk of boring everybody let me recall that this principle says, in its most naive form, that classical canonical coordinates and momenta are promoted to self-adjoint operators on some Hilbert space in the quantum theory with CCR commutator [ x&Hat;,p&Hat;]=ih .

One observes that this commutation relation may be exponentiated by defining U&Hat; (a)=exp(-iap&Hat;/h) and V&Hat; (a)=exp(-iax&Hat;/h) , which gives the Weyl form of the CCR: U&Hat; (a)V&Hat;(b)=e iab /hV&Hat;(b)U&Hat;(a).

One central idea of the LQG-like quantization is to modify the correspondence principle to the effect that instead of demanding x&Hat; and p&Hat; to be operators on some Hilbert space satisfying the CCR, one demands U&Hat; (a) and V&Hat; (a) to be operators on some Hilbert space and that the Weyl form of the CCR holds.

The crucial point is that there are representations of the Weyl algebra, namely those which are not weakly continuous (expectation values of V&Hat; (a) of U&Hat; (a) are not continuous in a ), which are not unitarily equivalent to that obtained by exponentiating operators x&Hat; and p&Hat; . LQG-like quantization wants to work with these ‘exotic’ representations of the Weyl algebra, that’s the program.

To my mind this program has the following problem, which, in different guises, has been discussed here a lot already: The problem is that classical equations of motion, classical constraints, the Schroedinger equation, etc., are usually expressed in terms of x and p . But now not both of these are available as operators x&Hat; and p&Hat; . So what is the quantization prescription then? Are we to do the exponentiation classically, in the Poisson algebra and then promote the result to an operator? By which rules to we specify the commutation relations of the resulting operators? Again by the classical theory?

I was hoping to find an answer to this important question in the above mentioned paper. It is page 14 of that paper where an aspect of this questions is discussed. There, the task is to find a Weyl-algebra analog of the definition of coherent states a&Hat; |ø æ&rangle;=æ|ø æ&rangle;, where a&Hat; is the lowering operator of ordinary 1d nonrelativistic QM. In this form this equation is not available in LQG-like QM, because there p&Hat; , which enters the definition of a&Hat; , is not defined. Therefore one has to find an exponentiated version of this equation.

Now comes the interesting point: The exponentiated version of the above equation in this paper is modeled after the respective exponentiated equation in in the usual Schroedinger quantization, using the Baker-Campbell-Hausdorff formula for the ordinary CCR algebra operators! (Please see page 14 of this paper for details.)

If this does not sound surprising,, recall how a similar step was done in Thomas Thiemann’s paper: There the question was how to represent the Virasoro constraints in exponentiated form, since the constraints themselves were not represented as operators in Thomas Thiemann’s LQG-like quantization of the string. Following the above paper by Ashtekar, Fairhurst and Willis one might have expected that this was done modeled after the usual quantum theory, which, as Jacques Distler has emphasized would still see the anomaly, of course. Instead, what Thomas Thiemann does in his paper is to use the classical Poisson-algebra of the exponentiated Virasoro constraints and represents this in terms of operators on some Hilbert space.

It seems to me that by modifying the usual correspondence principle (which incidentally also means giving up the path integral) one arives at a proposal for a new form of quantization which is not uniquely well defined. Of course a similar statement is true for the standard form of Schroedinger-like quantization, where ordering ambiguities in expressions like xp have to be dealt with. But the ambiguity in the LQG-like quantization seems to be much more severe.

If in Thomas Thiemann’s paper one were to follow the prescription indicated on page 14 of the Ashtekar,Fairhurst&Willis paper, one would find the anomaly. If one instead uses the classical algebra one misses it.

How are we supposed to deal with the ambiguities that arise as soon as the usual correspondence principle based on the CCR is replaced by one based on the Weyl-form of the CCR?

One way is indicated by Ashtekar,Fairhurst&Willis in their simple QM example: If one takes care that the usual relations are correctly translated to the new formalism (as is done on their page 14) then one finds the same results as in the usual quantum theory, essentially. In this case, however, one might wonder what the LQG-like formalism buys us.

The other way is to model the Weyl-CCR operator relations after the classical algebra, as done by Thomas Thiemann for the LQG string and in general in LQG for the spatial diffeomorphism constraints of gravity. This approach however has very little in common with what one usually calls ‘quantization’. And it is also doubtful that an argument as in Ashtekar,Fairhurst&Willis can recover the usual theory in this case, I think."

Too bad the math symbols don't copy and paste. Nor did the copy and paste work on the last line of text! Thanks to Jeff supplying them I have edited in the missing part of the last sentence.

Originally posted by marcus Great! Thanks. I will immediately paste it in here. I was put off when he refused to give me a link like that.

This approach however has very little in common with what one usually calls ‘quantization’. And it is also doubtful

You know what put's me off marcus? That you omitted the final few words, and since they were really the crucial part, it's impossible not to see that you did this on purpose to mislead. Here's the rest of the quote:

"his approach however has very little in common with what one usually calls ‘quantization’. And it is also doubtful that an argument as in Ashtekar,Fairhurst&Willis can recover the usual theory in this case, I think."

Translation, the special argument in the shadow states paper wil not extend to broad LQG. Probably not (I don't "know" that, but then I don't "know" very much). So what? If you are just cautioning into reading too much into the paper, consider me cautioned.